YAN Qing-you, XIONG Xi-wen. An Effcient and Stable Structure Preserving Algorithm for Computing the Eigenvalues of a Hamiltonian Matrix[J]. Applied Mathematics and Mechanics, 2002, 23(11): 1150-1168.
Citation: YAN Qing-you, XIONG Xi-wen. An Effcient and Stable Structure Preserving Algorithm for Computing the Eigenvalues of a Hamiltonian Matrix[J]. Applied Mathematics and Mechanics, 2002, 23(11): 1150-1168.

An Effcient and Stable Structure Preserving Algorithm for Computing the Eigenvalues of a Hamiltonian Matrix

  • Received Date: 2001-02-27
  • Rev Recd Date: 2002-06-28
  • Publish Date: 2002-11-15
  • An efficient and stable structure preserving algorithm,which is a variant of the QR like (SR)algorithm due to Bunse-Gerstner and Mehrmann,is presented for computing the eigenvalues and stable invariant subspaces of a Hamiltonian matrix.In the algorithm two strategies are employed,one of which is called dis-unstabilization technique and the other is preprocessing technique.Together with them,a socalled ratio-reduction equation and a backtrack technique are introduced to avoid the instability and breakdown in the original algorithm.It is shown that the new algorithm can overcome the instability and breakdown at low cost.Numerical results have demonstrated that the algorithm is stable and can compute the eigenvalues to very high accuracy.
  • loading
  • [1]
    Byers R. A Hamiltonian QR-algirithm[J]. SIAM J Sci Statist Comput,1986,7:212-229.
    [2]
    Bunse Gerstner A, Byers R, Mehrmann V. A chat of numerical methods for structured eigenvalue problems[J]. SIAM J Matrix Anal Appl,1992,13:419-453.
    [3]
    Bunse-Gerstner A, Mehrmann V. A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation[J]. IEEE Trans Automat Control,1986,31:1104-1113.
    [4]
    Hench J J, Laub A J. Numerical solution of the discrete-time periadic Riccati equation[J]. IEEE Trans Automat Control,1994,39:1197-1210.
    [5]
    Lin W W. A new method for computing the closed loop eigenvalues of a discrete-time algebraic Riccati equation[J]. Linear Algebra Appl,1987,96:157-180.
    [6]
    Lu L Z, Lin W W. An iterative algorithm for the solution of a discrete-time algebraic Riccati equation[J]. Linear Algebra Appl,1993,188/189:465-488.
    [7]
    Lin W W, Wang C. On computing stable Lagrangian subspaces of Hamiltonian martices and symplectic pencils[J]. SIAM J Matrix Anal Appl,1997,18:590-614.
    [8]
    Pappas C, Laub A J, Sandell N R. On the numerical solution of the discrete-time algebraic Reiccati equation[J]. IEEE Trans Autom Control,1980,25:631-641.
    [9]
    Patel R V. On computing the eigenvalues of a symplectic pencils[J]. Linear Algebra Appl,1993,188:591-611.
    [10]
    Patel R V, Lin Z, Misra P. Computation of stable invariant subspaces of Hamiltonian matrices[J]. SIAM J Matrix Anal Appl,1994,15:284-298.
    [11]
    Benner P, Mehrmann V, Xu H. A numerically stable, structure preserving method for computing the eigenvalues oy real Hamiltonian or symplectic pencils[J]. Numer Math,1998,78:329-358.
    [12]
    Bunse Gerstner A, Mehrmann V, Watkins D. An SR algorithm for Hamiltonian matrices, based on Gaussian elimination[J]. Methods Oper Res,1989,58:339-358.
    [13]
    Mehrmann V. A symplectic orthogonal method for single input or single output discrete time optimal quadrtic control problems[J]. SIAM J Matrix Anal Appl,1988,9:221-247.
    [14]
    Van Loan C. A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix[J]. Linear Algebra Appl,1984,16:233-251.
    [15]
    许波,刘征. Matab工程数学应用[M]. 北京:清华大学出版社,2000.
    [16]
    Golub G H, Van Loan C. Matrix Computations[M]. Baltimore: The Johns Hopkins University Press,1996.
    [17]
    Stewart G W. Introduction to Matrix Computations[M]. New York: Academic,1973.
    [18]
    Wilkinson J H. The Algebraic Eigenvalue Problem[M]. Clarendon: Oxford,1965.
    [19]
    Benner P, Fabender H. An implicity restarted symplectic lanczos method for the Hamiltonian eigenvalue problem[J]. Linear Algebra Appl,1997,263:75-111.
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (2685) PDF downloads(1479) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return